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Low Level RF Controls for SC Linacs, JLab April 24 - 27, 2001 M. Luong, A. Mosnier

Basic rules for the design of RF Controlsin High Intensity Proton Linacs

Particularities of proton linacs wrt electron linacs …� Non-zero synchronous phase

⇒ needs reactive beam-loading compensation

� Phase slippage (inside and outside cavities)

⇒ larger sensitivity to cavity field fluctuations

� Different Dynamic properties of a given cavity family + phase slippage

⇒ groups of multiple cavities driven by 1 common klystron

can only be used at sufficiently high energy

Basic rules


K

Phaseshifter

I Q

Ig Ib

-GQ(V Qc-V Qr)-GI (V Ic -V Ir )

I Q

Att.

Limiter

M.O.

VI c VQ c

M.O.

Measurement

Self-excited loopGenerator driven

Feedback loops

Generic RF control scheme(SEL or GDR)

Beam-loading compensationCavity detuning ∆ω

ω= −

R Q I0Vc

sinφs

VIc = Re(V c)= V c cos φc

VQc= Im(V c)= V c sin φc

Basic rules


Example of SEL scheme with analog control system implementedon SC cavity with non-relativistic beam (φb = 30°) = TTF capture cavity (Ein = 250 keV)

Main features :

• self-excited loop (which ensures the cavityfrequency tracking during the filling time)

• Cavity field controlled by means of I/Qmodulator (during filling and beam-on time)

• starting phase of the self oscillator fixed byinjection of a very low level signal

• klystron phase loop including a phase modulator to compensate any phase shift of the klystron

tests at DESY in February 1997 Eacc=12.5 MV/m, 6 mA

Amplitude & phase errors : ± 4 10-4 & ± 0.1°peak-to-peakfluctuations mainly from high frequency noise (uncorrelatedfrom cavity to cavity ⇒ small effect on beam energy spread)

0.6°

Basic rules


Basic Model for Gain-Stability Considerations

These 2 loops can be considered as independent for the stability discussion, having the sametransfer function. A simple representation can be found with several, but however realisticassumptions in the case of SC cavity :

2 Control loops for field stabilization in cavity : real part loop and imaginary part loop

• klystron and I/Q modulator bandwidths are much higher than that of the cavity,• closest harmonic mode (5π/6 for SNS and 8π/9 for TESLA about 800 kHz from the fundamental mode in both cases) is filtered out properly using HF pass-band filter and audio notch filter.

Simple model for analog feedback

For fLP = 400 kHz, K = 180, F0 = 1.3 GHz,and QL = 3 106, the phase margin fromthe model is 86°.

High gain and good phase margin are allowed for proportional analog feedback

TTF capt. Cav.

Basic rules

measurement


Gain-Delay Discussion in Proportional Digital Feedback Control

Open loop transfer function

cav

cavpdT

pdTlogana pp

e1e.K)p(H

ωω+

−=−

−

Z transform

( )( )dcav

dcav

T

T

digitalezz

e1K)z(H ω

ω

−

−

−−=

Roots of equation : 1 + Hdigital(z) = 0should lie inside the unity circle forstability of the closed loop.

L

0cav

cavd Q

fwith,

1KK

Ln1

T πωω

=

−=

(Assuming : sampling period = delay)

Limit of stability :

Without low-pass filter With low-pass filter for noise reduction

( )( ) dTLP�

dTLP�

dTcav�

dTcav�

ez

e1

ezz

e1KH(z) −

−

−

−

−−

−−=

dzczbza

)z(H1)z(H

)z(G 23 +++=

+=

3n2n1n3nn ydycybxay −−−− −−−=Recursion equation for time domain response

Closed loop discrete transfer function

Open loop transfer function

( )( )( )( ) ( )cb1Kd,ec

,eeb

,e1e1Ka

dTLPcav

dTLPdTcav

dTLPdTcav

++==

+−=

−−=

+−

−−

−−

ωω

ωω

ωω

Basic rules


Gain-Delay Discussion in Proportional Digital Feedback ControlResults for the case of SNS

0

5

10

15

20

0 40 80 120 160

wo filterLP filter (350kHz)

Open Loop Gain

frequency = 805 MHzQL = 7e5

STABLE

UNSTABLE

Without filter

With LP filterK = 25, Td = 5 µs ⇒ Phase margin = 40° w/o filter and 30° with filterTime response to a step excitation (on the right)

Stability could be more critical for digital feedback because of delay

Basic rules


Sampling of the measurement …

Bunch Trains are 945 ns spaced apart⇒ Systematic cavity voltage drop due to beam-loading

Better to sample field measurementJust before beginning of bunch train*

In order to avoid systematic extra-powerfrom feedback loopsEven in absence of any perturbation

* in case of digital system with delay,sampling time should be multipleof bunch train spacing

-2 10-3

-1 10-3

0 100

1 10-3

2 10-3

-1 10-6 0 1 10-6 2 10-6 3 10-6 4 10-6 5 10-6 6 10-6

∆V

c / V

c

time (s)

-0.2

-0.1

0

0.1

0.2

-1 10-6 0 1 10-6 2 10-6 3 10-6 4 10-6 5 10-6 6 10-6

∆P

/ P

(in

-pha

se)

time (s)

Fast feedback loops (G=100)1st medium-beta SNS cavityBeam current = 52 mAChopper duty factor = 68 %

continous measurement

sampled measurement

Basic rules


Ψφb I g

I tI b

Vc/R

Beam-loading compensationduring beam pulsedetuning angle such that cavity voltage looks “real”generator current and cavity voltage in phase φc=φg

steady-state regime with beam(“flat” Vc and φccurves)

during cavity fillingOne wants at fill end φc=φg whatever the detuningAutomatically satisfied with SEL schemeNot the case for a pure GDR scheme

Pg = Pgβ + 1( )2

4β(1+ Ycosφb)2 + (tanΨ + Ysinφb)2[ ]

beam - loading parameterY = RLIbVc

detuning angle tanΨ = 2QL∆ωω

tanΨ + Ysinφb = 0

0

0.2

0.4

0.6

0.8

1

1.2

-5

0

5

10

15

-2.5 10-4 0 100 2.5 10-4 5 10-4 7.5 10-4

time (s)

Vc / Vco

φ c (deg)

Vg = (1+ Ycosφb) Vc

Basic rules


Simulation code = PSTAB• initially developed for relativistic beams has been extended to low beta beams• can handle all major field error sources

(Lorentz forces, microphonics, input energy offsets, beam charge jitter, multiple cavities driven by one single power source, etc)

• includes feedback system and extra power calculation (+ delay for digital system)• with N cavities driven by a common klystron,

solves the 6xN coupled differential equations per power source :

- 3 differential equations per cavity for beam-cavity interactiononce the linac configuration has been defined (cavity types, number of cryomodules, design acceleratingfield and synchronous phase) a reference particle is launched through the linac in order to set the nominalphase of the field with respect to bunch at the entrance of all cavities

- 3xN equations per klystron for cavity fielddynamics of each resonator described by 2 first order differential equations,plus another one modelling dynamic cavity detuning by the Lorentz forcesbeam-loading is modelled by a cavity voltage drop during each bunch passage with a magnitude varyingfrom cavity to another (particle speed varies)

To minimize the needed RF power :1) Qex is set near the optimal coupling " 7.105 for SNS2) Cavity is detuned to compensate the reactive beam-loading due φs ? 0 and fine tuning adjustement for minimizing Lorentz force effects

Simulations


Linac Descr iption*****************Starting Energy (M eV) : 185.6Nb sect ors : 2 *** Sector w ith Cavity Type 1Acc gradient (M V/m ) : 10.11Nb Cav / cryom odule : 3Nb cryom odules : 11Cavity spaci ng (m ) : 0.51Inter-cryom drift (m ) : 2.7752Nb cavi ties / Klyst ron : 1Beam phase standar d (deg) : -22. *** Sector w ith Cavity Type 2Acc gradient (M V/m ) : 12.56Nb Cav / cryom odule : 4Nb cryom odules : 15Cavity spaci ng (m ) : 0.48Inter-cryom drift (m ) : 2.866Nb cavi ties / Klyst ron : 1Beam phase standar d (deg) : -22. Total Nb cavi ties = 93 Total Nb klyst rons = 93 Linac Phasing************* Sector 0 Ending Energy = 185.600 Sector 1 Ending Energy = 388.526 Sector 2 Ending Energy = 974.595

Beam param eters***************Peak Beam cur rent (m A) : 52.Current fluct uation (% ) : 0.Harm onic (Frf/Fb) : 2.Nb of bunches / train : 260Nb of m issi ng bunches : 120Nb of bunch trains : 1060

Example of PSTAB simulationswith SNS parameters

-240

-220

-200

-180

-160

-140

-120

0 20 40 60 80 100

∆f b(H

z)

cavity index

Cavity detuning for beam-loading compensation

2 systems were tested :analog (G=100) & digital (G=25 delay 4.7us)

Simulations


0

2

4

6

8

10

12

0 20 40 60 80

in-phaseout-of-phase

∆ P/P

(%

)

klystron index0

2

4

6

8

10

12

0 20 40 60 80


∆ P/P

(%

)

klystron index

analog

digital

Simulations

⇒ Lorentz forces effectsIn order to decrease RF power ⇒ pre-detuning of the cavity (fres = fope at approximately half the beam pulse)Total detuning = Sum of detunings for Lorentz forces + beam-loading compensation

-2

-1

0

1

2

-0.02

-0.01

0

0.01

0.02

0 0.0002 0.0004 0.0006 0.0008 0.001

Lorentz forces only ( 8 - 4 Hz / [MV/m] 2 )

∆φ (

deg)

∆E

(MeV

)

time (s)-2

-1

0

1

2

-0.02

-0.01

0

0.01

0.02

0 0.0002 0.0004 0.0006 0.0008 0.001

Lorentz forces only ( 8 - 4 Hz / [MV/m] 2 )

∆φ (

deg)

∆E

(MeV

)

time (s)


⇒ Additional microphonics effectsWith mechanical vibrations, feedback loops closed during the filling time, following pre-determined amplitudeand phase laws, to ensure min RF power during beam pulse

0

5

10

15

20

0 20 40 60 80


∆ P/P

(%

)

klystron index0

5

10

15

20

0 20 40 60 80


∆ P/P

(%

)

klystron index

analog

digital

Simulations

-4

-2

0

2

4

-0.1

-0.05

0

0.05

0.1

0 0.0002 0.0004 0.0006 0.0008 0.001

Lorentz + µphonics (100 Hz)

∆φ (

deg)

∆E

(MeV

)

time (s)-4

-2

0

2

4

-0.1

-0.05

0

0.05

0.1

0 0.0002 0.0004 0.0006 0.0008 0.001

Lorentz + µphonics (100 Hz)

∆φ (

deg)

∆E

(MeV

)

time (s)


⇒ Additional current fluctuation effectWith random bunch charge fluctuation, the feedback system prevents from dramatic cumulative effects ofseveral consecutive bunch charge errors

0

2

4

6

8

10

12

0 20 40 60 80


∆ P/P

(%

)

klystron index0

2

4

6

8

10

12

0 20 40 60 80


∆ P/P

(%

)

klystron index

analog

digital

Simulations

-2

-1

0

1

2

-0.1

-0.05

0

0.05

0.1

0 0.0002 0.0004 0.0006 0.0008 0.001

Lorentz + charge fluctuation (5%)

∆φ (

deg)

∆E

(MeV

)

time (s)-2

-1

0

1

2

-0.1

-0.05

0

0.05

0.1

0 0.0002 0.0004 0.0006 0.0008 0.001


∆φ (

deg)

∆E

(MeV

)

time (s)


-2

-1

0

1

2

-8 -6 -4 -2 0 2 4 6 8

input beam offset ( ± 1 MeV ± 1 deg )with constant fields

∆E (

MeV

)

∆φ (deg)

6°1.

3 M

eV

⇒ input beam offset

With random energy and phase errors of the input beam ( ± 1 MeV ± 1 deg ) and

assuming no fluctuations of the cavity fields

⇒ the final energy and phase can not be smaller than 1.3 MeV and 6 deg .

Simulations


Simulations

SEL scheme analog system digital system (4.7 µs delay)

∆E max (keV) ∆φ max (deg) ∆P/PI (%) ∆P/PQ (%)

Lorentz forces only 4.8 0.51 0.2 9.0

8 - 4 Hz/[MV/m]2 17.1 1.61 1.2 11.2

with µphonics A=100 Hz 17.8 0.53 0.3 16.1

Freq = 100 Hz 70.4 1.68 1.3 17.9


Freq = 1000 Hz 158.5 1.78 1.3 16.9

with tuning drift 12.3 0.54 0.3 16.6

± 100 Hz (random) 57.6 1.75 1.2 18.3

with charge fluctuation 46.5 0.59 4.2 9.1

± 5 % (random) 96.3 1.68 3.3 11.3

with input beam offset 1303.7 6.14 5.9 11.6

± 1 MeV ± 1 deg (random) 1329.0 6.76 4.1 11.1

multi-perturbation* 1321.0 6.25 7.8 18.5

1405.5 7.10 5.7 18.0

* Lorentz forces + µphonics (Freq=100 Hz) + charge fluctuation + input beam offset


Simulations

GDR scheme analog system digital system (4.7 µs delay)


Lorentz forces only 5.7 0.51 0.5 8.7

8 - 4 Hz/[MV/m]2 27.9 1.51 2.0 9.4


Freq = 100 Hz 96.3 1.56 3.1 15.7


Freq = 1000 Hz 165.3 1.60 3.2 14.7

with tuning drift 12.6 0.54 0.8 16.0

± 100 Hz (random) 48.4 1.63 3.2 16.4

with charge fluctuation 46.6 0.59 4.3 8.8

± 5 % (random) 108.3 1.58 4.2 9.5

with input beam offset 1304.1 6.14 6.2 11.2

± 1 MeV ± 1 deg (random) 1332.5 6.70 5.1 9.4

multi-perturbation* 1322.1 6.25 8.2 17.9

1407.1 7.05 7.4 15.6



Conclusion

� SEL scheme vs GDR scheme

SEL requires smaller in-phase power and larger out-of-phase power(SEL keeps naturally the amplitude but shifts the phase)

� Analog vs digital

generally digital with delay increases extra-power and errorsunless total delay time " 1 µs

for a random input beam offset :extra-power is slightly smaller but final errors are larger(because feedback can't follow with sudden changes)

Simulations


Multiple cavities per klystron

� With relativistic electron beams, multiple cavities powered by a single powersource easily controlled by the vector sum of the cavity voltages

� With proton beams, even when the vector sum kept perfectly constantindividual cavity voltages can fluctuate with large amplitudes(phase slippage + change of dynamic behaviour of low-β cavities as energy � )

We could however envisageto feed individually the cavities at the low energy part of the SC linac &to feed groups of cavities by common klystrons at the high energy part(lower phase slippages + closer dynamic cavity properties)

For example, groups of 4 cavities only for high-β cavities in SNS LinacSimulations with spreads in� Lorentz force detuning K 20 % 2 Hz / [MV/m]2

� mechanical time constant τm 20 %� cavity coupling Qex 20 %

Multi-cavity


Multi-cavity

-0.04

-0.02

0

0.02

0.04

-0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001

1st high-beta cryomodule

Lorentz Forces ( 2 Hz / [MV/m] 2 )∆V

/V

time (s)-0.04

-0.02

0

0.02

0.04

-0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001

1st high-beta cryomoduleLorentz Forces + spread in K, τ

m, Q

ex∆V

/V

time (s)

Ex. Cavity voltages of 1st high-beta cryomodule (analog system G = 100)


⇒ Lorentz forces & charge fluctuation (5%) effectsBunch energy deviations at beginning mainly induced by Qex spreadExtra-power at end mainly induced by current fluctuations

0

2

4

6

8

10

0 10 20 30 40


∆ P/P

(%

)

klystron index

Multi-cavity

-1

-0.5

0

0.5

1

-0.5

-0.25

0

0.25

0.5

0 0.0002 0.0004 0.0006 0.0008 0.001


∆φ (

deg)

∆E

(MeV

)

time (s)


SEL scheme analog system


Lorentz forces only 418 0.26 1.6 2.8

4 - 2 Hz/[MV/m]2

with µphonics A=100 Hz 688 0.93 2.5 6.4

Freq = 100 Hz

with charge fluctuation 418 0.31 8.9 2.9

± 5 % (random)

with input beam offset 1600 6.1 11.9 5.8

± 1 MeV ± 1 deg (random)

multi-perturbation* 1875 6.4 16.3 8.9


Multi-cavity

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